Background Information on the LPN

Background to the Learning Pathway for Number [Ch-head]

Introduction [A-head]

The Learning Pathway for Number (LPN) gives an overview of learners’ growth in understanding of whole numbers in the early years of primary school. It provides insight into how this process evolves and can be stimulated. The LPN is linked to the National Curriculum Statement (NCS) and supports teachers in the implementation of the NCS by presenting a framework that describes the knowledge base of how learners develop number concepts and number skills from preschool to Grade 4.

Learners progress through several distinct but related stages in this learning process. Drawing on developmental milestones, the LPN distinguishes between four stages of numerical development which broadly describe what can be expected of learners’ number development within and across the lower primary grades. Each stage describes specific qualities of understanding that learners demonstrate and which can be encouraged and built on in teaching– learning situations. These stages are not fixed and, moreover, it is not always possible to identify exactly what understanding learners have. The stages are not meant for classifying learners. Rather, they are meant as a guide to understanding their thinking and as a reference for making decisions about further teaching.

The transition from informal counting-based strategies to formal operations underpins the structure of this book. Descriptions of each stage outline the main features and aspects of learners’ early number knowledge and provide insight into their thinking and a basis for observing and recording their knowledge.

13 Links to the National Curriculum Statement [A-head]

The LPN outlines a progressive sequence of learners’ number development. In addition, it highlights the ‘stepping stones’ that learners will pass on their way to reaching the NCS Mathematics outcomes related to number (Learning Outcomes 1 and 2). The knowledge and skills needed to achieve the NCS Assessment Standards related to number at the end of each grade in the Foundation Phase are presented in a table on the next page.

The Assessment Standards for LO1 and LO2 give a clear description of the minimum requirements for number knowledge and skills in each grade. However, the NCS does not explain which teaching approaches and learning environments can facilitate the achievement of these exit points. Neither does it describe the developmental sequence that learners follow in learning to count and calculate, nor what teachers should do to make this development possible.

The LPN aims to fill this gap. It provides an overview of how number knowledge, number sense, and mental and written calculation up to 1000 are developed and relate to each other within and across the Foundation Phase Mathematics Learning Outcomes. The LPN is designed to guide the teaching and learning of mathematics along the route to achieving the NCS Assessment Standards for the lower primary grades (Grade R to 4).

14 NCS Number-Related Knowledge and Skills: LOs 1 and 2 Skills Grade R Grade 1 Grade 2 Grade 3 Counts To at least 10 To at least 34 To at least 100 To at least 1000 objects reliably Says and In familiar uses numbers contexts Count forward ▫ In 1s from 0 to ▫ 1s from 0 to ▫ As Grade 2 and 100 200 ▫ 20s, 25s, 50s backwards ▫ In 10s from 0 to ▫ 10s from 0 to and 100s 100 200 between 0 and ▫ 5s from 0 to 1 000 200 ▫ 2s from 0 to 200 Recognise Knows the Knows and reads Knows and reads Knows number number number names number symbols number symbols names from 1 to names and and symbols for from 1 to at least from 1 to at least at least 10 in the symbols 1 to 10 100 200 mother tongue (if not the language of learning and teaching) and one other local language Represent Writes number Writes number Writes number numbers names and names and names and symbols from 1 to symbols from 1 symbols from 1 at least 34 to at least 100 to at least 1000 Orders, Collections of Whole numbers to ▫ whole numbers ▫ whole numbers describes and objects using at least 2-digit to at least 2- to at least 3- compares the words numbers digit numbers digit numbers ‘more’, ‘less’ ▫ common ▫ common and ‘equal’ fractions fractions including including halves and halves, quarters quarters and thirds Recognises, To at least 100 To at least 200 To at least 1 000 describes, copies and extends simple number sequences Recognises Of digits in whole Of digits in whole place value numbers to at numbers to at least 2-digit least 3-digit numbers numbers Solves money Totals and change Totals and Totals and problems in rands and cents change in rands change in rands and cents and cents, including converting between rands and cents

15 Solves and Equal sharing Equal sharing and Equal sharing Equal sharing explains and grouping grouping with and grouping and and grouping solutions to with whole whole numbers to that lead to and that lead to practical numbers of at at least 34 and solutions that solutions that problems least 10 and with solutions that also include also include with solutions include unitary fractions unitary and that include remainders (e.g. ¼) nonunitary remainders fractions (e.g. ¼, ¾) Solves Single-digit verbally stated numbers with problems solutions to at least 10 Estimation Estimates Estimates Estimates Written ▫ addition and ▫ addition and ▫ addition and calculation subtraction with subtraction of subtraction of using whole numbers whole numbers whole numbers appropriate and solutions to with at least 2 with at least 3 symbols at least 34 digits digits ▫ repeated ▫ multiplication of ▫ multiplication addition with whole 1-digit by of at least whole numbers 1-digit numbers whole 2-digit and with with solutions by 1-digit solutions to at to at least 50 numbers lease 34 ▫ division of at least whole 2- digit by 1-digit numbers

Mental addition and ▫ addition and ▫ addition and calculation subtraction for subtraction for subtraction for numbers to at numbers to at numbers to at least 10 least 20 least 50 ▫ multiplication of ▫ multiplication whole numbers of whole with solutions numbers with to at least 20 solutions to at least 50 Calculation ▫ building up ▫ building up and ▫ building up and ▫ building up and techniques and breaking breaking down breaking down breaking down down numbers numbers numbers numbers to at ▫ doubling and ▫ doubling and ▫ doubling and least 10 halving halving halving ▫ doubling and ▫ using concrete ▫ using concrete ▫ number-lines halving to at apparatus (e.g. apparatus (e.g. ▫ rounding off in least 10 counters); counters) tens ▫ using ▫ number-lines ▫ number-lines concrete apparatus (e.g. counters)

16 A guide for teaching number [A-head]

The LPN is a tool for instructional decision making – it is not a detailed recipe for day-to-day teaching. Instead, it gives teachers a conceptual educational map of number development. It gives an overview of the long-term process of learning number. The theoretical framework describes the sequence of development and illustrates how the different developmental stages are related to one another. Typical performances at each stage of number development are described, together with suggestions for activities that can be used for teaching counting and calculation in the Foundation Phase.

This LPN should not be seen as isolated from other pathways for learning and teaching the mathematical strands of fractions, measurement, shape and space. There are many links to other Learning Outcomes and Learning Areas which can provide a rich context for exploring number. For example, a shape and space activity involving blocks and a mirror to explore symmetry could include discussions about odd and even numbers as well as halving and doubling.

Bear in mind that learners’ mathematical development cannot be separated from their general cognitive, physical, emotional and social development. These developments are inseparable but, at the same time, are distinguishable from one another. By looking at the developments separately, however, we get a more in-depth view of a learner’s development. This enables one to see how several developments are related and integrated.

Although the term ‘learning pathway’ suggests a linear and continuous sequence, it should not be considered as such. First of all, learning processes do not develop in the same way for every learner. In addition, learning processes do not even develop the same way during each stage. For example, learners from preschool to Grade 1 will benefit most from games and integrated activity-based learning environments. On the other hand, number activities become a focus in themselves in the higher grades where

17 mathematics education is organised in planned and focused lessons. Of course, this need not be true for all learners. Teaching at each stage will finally be determined by the individual’s number development and level of understanding.

Links with teaching principles of Realistic Mathematics Education [A-head]

The method of teaching mathematics that is intended in the LPN is aligned with the teaching principles of Realistic Mathematics Education. These are:  the context principle  the level principle  the activity principle  the interaction principle  the guidance principle.

The context principle means that the teaching of mathematics is based on the broadly recognised assumption that the learning of mathematics requires contexts that are meaningful for learners. Significant contexts and the problems that arise from them trigger the development of mathematical concepts and tools because there is a need for learners to solve these problems. The level principle is strongly related to the context principle. It comprises the idea of scaffolding and shifts in levels of understanding. This means that the teaching is aimed at a transition from concrete and informal ways of working to abstract and formal ways of working. In other words, context-connected strategies gradually evolve into more general and abstract strategies. Models,

 Treffers, A. (1987). Three Dimensions. A Model of Goal and Theory Description in Mathematics Instruction – the Wiskobas Project. Dordrecht: Reidel Publishing Company. See also, Van den Heuvel-Panhuizen, M. (2001). Realistic Mathematics Education in the Netherlands. In J. Anghileri (Ed.), Principles and Practices in Arithmetic Teaching: Innovative Approaches for the Primary Classroom (pp. 49-63). Buckingham/Philadelphia, United Kingdom: Open University Press.

18 such as the number line, that emerge from contexts, such as a chain of beads, serve as a bridge to reaching these more general and abstract levels of understanding.

The activity principle is grounded in Freudenthal’s basic idea of ‘mathematics as a human activity’. This means that learners should not be seen as receivers of ready-made mathematics, but should play an active role in the learning–teaching process. This approach is also in agreement with the generally acknowledged constructivist view on learning.

The interaction principle is related to the highly valued and necessary social aspect of learning. By listening to what others have figured out and by discussing these findings, learners can get ideas for improving their strategies. Moreover, the interaction can evoke reflection, which enables learners to shift to a higher level of understanding. The significant role of interaction has important consequences for classroom organisation. The interaction principle implies a preference for whole-class teaching rather than the splitting of classes into small groups that are taught separately, However, this does not mean that the whole class is developing at the same pace and that every learner is following the same track or reaching the same level of development at the same time. On the contrary, learners are considered as individuals that develop at their own pace. Therefore, whole-class teaching should be linked to teaching activities that are adapted to individual learner’s needs by providing problems that can be solved using strategies based on different levels of understanding. Finally, the guidance principle is also heavily based on Freudenthal’s view that mathematics education should give students a ‘guided’ opportunity to ‘reinvent’ mathematics. This implies that both the teachers and the educational programmes play a crucial role in how learners acquire knowledge. They should steer the learning process, not in a fixed way by

 Freudenthal, H. (1973). Mathematics as an Educational Task. Dordrecht, The Netherlands: Reidel Publishing Company. Freudenthal, H. (1991). Revisiting Mathematics Education. China Lectures. Dordrecht, The Netherlands: Kluwer Academic Publishers.

19 demonstrating what has to be learnt, but rather by asking questions and providing learners with problems that provoke the need or provide the room for them to construct mathematical insights and tools. Therefore, teachers must be able to anticipate learners’ understanding and skills that are just coming into view. The LPN provides teachers with knowledge about this longitudinal pathway.

Implementation of the Learning Pathway for Number [B-head]

Successful implementation of the LPN relies on school teams and teachers actualising the LPN in real learning environments in their classrooms. The conceptual educational map for the learning of number in the Foundation Phase that is provided by the LPN will guide them in this respect.

The LPN helps teachers to: ─ pinpoint the current understanding of individual learners and recognise the differences in progress as a stage along the pathway ─ understand the next step in the learning process and know how to select challenging and supportive tasks that stimulate learners’ development ─ identify learners’ strategies, give them adequate feedback, and choose appropriate didactic materials and models ─ take a broader perspective and be able to plan an instructional sequence that addresses the needs of all learners across a wide spectrum of number development ─ acquire a common language to discuss the teaching and learning of number with their colleagues.